I understand "to fall" and its use in figurative speech, but is the same word used in Danish to indicate, say, someone falling on their arse?
Yes. "falde" is used both literally and in figurative speech.
Min opmærksomhed falder nu (my attention is decreasing now) Drengen faldt og slog sig (the boy fell and hurt himself) Han faldt på knæ (he fell to his knees) Hvis du ikke passer på, falder du (if you are not careful, you will fall) Jeg hørte noget falde på gulvet (I heard something fall on the floor)
It stands here that "den" could also mean "that" but still I got it marked as incorrect when I wrote "he fell for that beautiful girl"
'den smukke pige' = the beautiful girl. So the definite meaning is located in the 'den'.
In English, "fall for" has the following meanings (from thefreedictionary): 1. To feel love for someone; be in love with someone: I fell for you the first time I saw you in the park. They immediately fell for each other. 2. To be deceived or swindled by something: The gullible dupe fell for the con artist's scheme and lost $200,000.
I love this about studying something significantly related to English. I also felt very happy to learn that give up is virtually identical this morning. :) Jeg elsker de skandinaviske sprog.